3,128 reputation
729
bio website combinatorialgames.wordpress.…
location United States
age 27
visits member for 2 years, 9 months
seen Nov 7 at 11:55

I have an amateur interest in combinatorial game theory and rarely update a blog with some basic exposition on the subject (see website).

If you need to contact me, use the e-mail address at this link.


Oct
13
comment countable subset of surreal games
If you want to go beyond the surreals and use division, you're doomed because division doesn't generally make sense (and to the extent that it would inherit sense from game addition, it's not well defined: *+*=0, does that mean *=0/2?). I don't think you'll like this answer, but finite exact representations of surreals with infinite sign expansions can be written with standard fraction and ordinal notation. "1/3" is a finite exact representation of a surreal with an infinite sign expansion. "$1/\omega +\omega_1/\omega_2$ is a finite exact representation of an uncountably long sign expansion.
Oct
13
comment countable subset of surreal games
To address your last question first, division doesn't generally work out for non-surreal games, and to the extent that it does, it's not unique: Should * be considered 0/2 since *+*=0? You are likely doomed if you're looking to go beyond the surreals. -- The nice way to use division over some surreals with finite representations is to use the standard notation for rational numbers and ordinals, etc. $1/\omega+1/(1+1/\omega_1)$ is a finite string representing a surreal number that's not a rational. $1/3$ is a finite representationing a surreal number with infinite sign expansion. etc.
Sep
30
comment What does a complex root signify?
This is just a random example, but the complex roots of $x^2+1=0$ are in some sense the reason the radius of convergence for the series expansion at $x=0$ of $1/(x^2+1)$ is $1$.
Sep
26
comment countable subset of surreal games
What do you mean by the phrases "those which are countable", "computable game", or "the countable subset of all games in a finite number of steps"? If you have to output countably many things, and outputting one thing is a step, you're not going to be able to do it in finitely many steps. You could take the computable sign expansions (analogous to computable binary expansions of reals), but even if that's what you want, I'm not sure what question you're asking about them.
Sep
24
comment Is there a more general concept than position and space?
"Draw 3 maps, one for each object" strongly brings to mind the concept of a "manifold", where local maps (called "charts" so you don't confuse it with the "function" meaning of the word "map") combine together to give information about the whole space.
Sep
23
comment CGT: value of sum game is sum of values of games
@HennoBrandsma I respectfully disagree. Addition of games yields a game in which players can move in either component. A priori, there's no reason to think that the naming convention for, say, the games which are "numbers" would work out so that all standard addition facts are preserved.
Sep
17
comment Which is greater, $20 \uparrow\uparrow\uparrow\uparrow 20$ or $4 \uparrow\uparrow\uparrow\uparrow\uparrow 4$?
I realized later that since $2^{10}$ is about $10^3$ (and less than $10^4$ for sure), you can get ballpark estimates/bounds on $20^{20}$ and $4^{256}$ without a calculator. $20^{20}$ is about $10^{26}$ and less than $10^{28}$, but $4^{256}$ is over $10^{153}\gg10^{28} $.
Sep
7
comment Construction of an infinite number type and other ideas
@11dim I still don't know if you're asking a question, or commenting on my answer, or just trying to advertise your ideas in the comments to an answer, which is probably not the best place for them.
Jul
19
comment Integration problem $\displaystyle \int \frac{dx}{x(x^3+8)}$
Can you find a zero of $x^3+8$? That can help you factor the denominator.
Jul
19
comment On which occasions will the intelligent layperson fail to recognize a mathematics problem?
There are problems with which mathematics can help, like "dividing up rent among a group of people", but given a precise phrasing of the corresponding mathematical problem, I think it would sound much more like mathematics. Would things like that count?
Jul
19
comment A question about limit 22
@BarryCipra Whoops! Well that's boring. Thanks; I'll edit.
May
26
comment How exactly is $i=\sqrt{-1}$ related to $\mathbb{C}$ being a closed algebraic field?
It's important to note (and the links clarify this) that the Quaternions are not what you get when you have the same goal (algebraic closure) but happen to be working with matrices, but rather something reminiscent of the complex numbers you can get when you change/weaken the goal.
May
26
comment How exactly is $i=\sqrt{-1}$ related to $\mathbb{C}$ being a closed algebraic field?
@Nikos $\sqrt\pi$ is the number that you square to get $\pi$. Every positive number has a positive square root: If you believe it for rationals, just take the limit of a sequence of rational numbers whose square is not quite big enough, but whose squares tend to the number in question.
May
26
comment Are there non-zero combinatorial games of odd order?
The proof of "no (nonzero) games of odd order" is too long to fairly reproduce here. I suppose it's worth mentioning that the key result (which is not so easy to prove) is that if $G$ has finite order and birthday $n$, then $2^nG=0$.
Apr
19
comment Are there 3 trig functions or are there 6 trig functions?
This appears to assume cosine is positive.
Mar
26
comment Necessary/sufficient conditions for an infinite product to be exactly equal to $1$
@pbs it was a typo since the text said equal to zero.
Mar
25
comment Necessary/sufficient conditions for an infinite product to be exactly equal to $1$
@sabyasachi yes, but the sequence 3,1,5,1,7,1,... diverges due to oscillation.
Mar
25
comment Necessary/sufficient conditions for an infinite product to be exactly equal to $1$
@sabyasachi I don't think those terms have product limit 1, but Daniel ' s example is fine.
Mar
25
comment Necessary/sufficient conditions for an infinite product to be exactly equal to $1$
The limit of the terms better be 1, but if you're strictly monotonically increasing or decreasing, then all terms are on one side of 1, which means the product won't be 1.
Mar
21
comment $\max_{y} \min_{x} f(x,y)$ as motif for exploring mathematics
@alex.jordan for every fixed $ y $, the function of a single variable $ g_y (x)=f (x, y) $ may have a minimum value, but different $ y $ s will give different minima. You can collect them all up into a function of $ y$ called $\min_x f (x, y) $. Since the min of $x^2/2-\pi x $ is $-\pi^2/2$, and similarly if we replace $\pi $ by any arbitrary number $ y $, lilinjn's expression makes sense